On polynomial reducibility of word problem under embedding of recursively presented groups in finitely presented groups

Valiev, M K

doi:10.1007/3-540-07389-2_229

“…It should be pointed out that the present work is quite different in content from the work of Trakhtenbrot [15] and Valiev [16,17] on the efficient reduction of word problems to halting problems. Both these authors are concerned with efficiently deciding, by ordinary computation, whether a given word W = 1.…”

mentioning

confidence: 45%

Efficient computation in groups and simplicial complexes

Stillwell¹

1983

Trans. Amer. Math. Soc.

0

View full text Add to dashboard Cite

Abstract. Using HNN extensions of the Boone-Britton group, a group E is obtained which simulates Turing machine computation in linear space and cubic time. Space in E is measured by the length of words, and time by the number of substitutions of defining relators and conjugations by generators required to convert one word to another. The space bound is used to derive a PSPACE-complete problem for a topological model of computation previously used to characterize NP-completeness and RE-completeness.Introduction. The ability of mathematical systems to simulate computation has often been used to prove unsolvability results. The first, and most instructive, example was Post's simulation of Turing machines by finitely presented semigroups [10]. For each deterministic Turing machine M, Post constructs a semigroup T(M) on generators we shall call qa, sb, where a and A range over certain finite sets. An instantaneous description (ID for short) of the state of computation at any moment can be written as a word 2 in these generators, there is a special qa generator called q, and the defining relations of T(M) are such that 2 = q is derivable if and only if the ID 2 leads M to halt. Thus the halting problem for M is reduced to the word problem for T(M), and by choosing an M with unsolvable halting problem Post proved the unsolvability of the word problem for semigroups.The simulation of M by T(M) shows that "semigroups can compute". What is interesting from the viewpoint of computational complexity is that the derivation in T(M) which reflects a given computation of M has length bounded by a linear function of the length of computation. Thus "time" in T(M) (measured by the

show abstract

“…(Added June 6, 1982). It should be pointed out that the present work is quite different in content from the work of Trakhtenbrot [15] and Valiev [16,17] on the efficient reduction of word problems to halting problems. Both these authors are concerned with efficiently deciding, by ordinary computation, whether a given word W = 1.…”

mentioning

confidence: 73%

Efficient Computation in Groups and Simplicial Complexes

Stillwell¹

1983

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

Abstract. Using HNN extensions of the Boone-Britton group, a group E is obtained which simulates Turing machine computation in linear space and cubic time. Space in E is measured by the length of words, and time by the number of substitutions of defining relators and conjugations by generators required to convert one word to another. The space bound is used to derive a PSPACE-complete problem for a topological model of computation previously used to characterize NP-completeness and RE-completeness.Introduction. The ability of mathematical systems to simulate computation has often been used to prove unsolvability results. The first, and most instructive, example was Post's simulation of Turing machines by finitely presented semigroups [10]. For each deterministic Turing machine M, Post constructs a semigroup T(M) on generators we shall call qa, sb, where a and A range over certain finite sets. An instantaneous description (ID for short) of the state of computation at any moment can be written as a word 2 in these generators, there is a special qa generator called q, and the defining relations of T(M) are such that 2 = q is derivable if and only if the ID 2 leads M to halt. Thus the halting problem for M is reduced to the word problem for T(M), and by choosing an M with unsolvable halting problem Post proved the unsolvability of the word problem for semigroups.The simulation of M by T(M) shows that "semigroups can compute". What is interesting from the viewpoint of computational complexity is that the derivation in T(M) which reflects a given computation of M has length bounded by a linear function of the length of computation. Thus "time" in T(M) (measured by the

show abstract

“…First results have been obtained by Clapham [51] and Valiev [231] (see [189] for the history of these results): they proved that the solvability (even r.e. degree) of the word problem and the level in the polynomial hierarchy of the word problem is preserved under some versions of Higman embedding.…”

Section: Theorem 51 (Higman [122]) the Word Problem In A Finitely Gementioning

confidence: 94%

“…After [122], there were several results showing that embedding into a finitely presented group can preserve or even improve the algorithmic properties of the group. First results have been obtained by Clapham [51] and Valiev [231] (see [189] for the history of these results): they proved that the solvability (even r.e. degree) of the word problem and the level in the polynomial hierarchy of the word problem is preserved under some versions of Higman embedding.…”

Section: 1mentioning

confidence: 99%